New symmetric group actions make FQSym* and PQSym* into invariant spaces, and their r-parameter deformations give a nested chain of Hopf subalgebras with explicit bases and Hilbert series.
The Hopf algebras of symmet ric functions and quasi-symmetric functions in non-commutative variables a re free and co-free
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The free and parking quasi-symmetrizing actions
New symmetric group actions make FQSym* and PQSym* into invariant spaces, and their r-parameter deformations give a nested chain of Hopf subalgebras with explicit bases and Hilbert series.